# Solving for Bending Stress and Deflection in a Cantilever Plate

• FEAnalyst
In summary: These boundary conditions ensure that the strains in the y direction are zero at the constrained edge.In summary, the conversation discusses the difficulty of finding formulas for a cantilever plate with one edge clamped and all other edges free under uniformly distributed load. The problem is complex due to the lateral constraint at the clamping, but a solution can be found in a Polish book using equations for maximum bending stress and deflection. The results from this solution are in good agreement with finite element analysis. The edge boundary conditions are provided, ensuring that the strains in the y direction are zero at the constrained edge.
FEAnalyst
TL;DR Summary
How to calculate stress and deflection of a cantilever plate analytically ?
Hi,

books such as „Roark’s Formulas for Stress and Strain” or Timoshenko’s „Theory of plates and shells” provide formulas for maximum bending stress and deflection for many cases of rectangular plates. However, I can’t find a simple case of a cantilever plate (one edge clamped, all other edges free) subjected to UDL anywhere. The closest example is a plate with one edge clamped and all other edges simply supported (from Timoshenko’s book) but it’s not the same. There must be a way to solve this problem of a shelf-like plate analytically. Do you know where I can find appropriate formulas ?

UDL??

This is a pretty tricky problem because of the lateral constraint of the plate at the clamping. If it were allowed to slide laterally, that would make it much easier. I'm sure that, somewhere in the literature, this problem (more complex version) has been solved, but I have never researched it.

Dr.D said:
UDL??

What I mean is uniformly distributed load.

Chestermiller said:
This is a pretty tricky problem because of the lateral constraint of the plate at the clamping. If it were allowed to slide laterally, that would make it much easier. I'm sure that, somewhere in the literature, this problem (more complex version) has been solved, but I have never researched it.

Thanks. I noticed that this problem is much more complex than it seems. I know that such plate can be treated as a beam but it's not very accurate approach. Maybe it's possible to solve this case using trigonometric series or other complex method but I'm not sure how to do it.

I've found a solution in a Polish book "Collection of tasks on strength of materials" by Banasiak, Grossman and Trombski. This book features a derivation and the final equations are: $$\displaystyle{ \sigma_{x \ max}=\frac{3qa^{2}}{h^{2}}}$$ $$\displaystyle{ \sigma_{y \ max}=\nu \cdot \frac{3qa^{2}}{h^{2}}}$$ $$\displaystyle{ \sigma_{vM \ max}=\frac{3qa^{2}}{h^{2}} \cdot \sqrt{1+ \nu^{2}- \nu}}$$ $$\displaystyle{ w_{max}=\frac{3 (1- \nu^{2})qa^{4}}{2 E h^{3}}}$$
where: ##q## - magnitude of uniformly distributed load, ##a## - length of the plate (from fixed to free end), ##h## - thickness of the plate.
The results are in very good agreement with FEA (much better than when the plate is treated as a beam).

FEAnalyst said:
I've found a solution in a Polish book "Collection of tasks on strength of materials" by Banasiak, Grossman and Trombski. This book features a derivation and the final equations are: $$\displaystyle{ \sigma_{x \ max}=\frac{3qa^{2}}{h^{2}}}$$ $$\displaystyle{ \sigma_{y \ max}=\nu \cdot \frac{3qa^{2}}{h^{2}}}$$ $$\displaystyle{ \sigma_{vM \ max}=\frac{3qa^{2}}{h^{2}} \cdot \sqrt{1+ \nu^{2}- \nu}}$$ $$\displaystyle{ w_{max}=\frac{3 (1- \nu^{2})qa^{4}}{2 E h^{3}}}$$
where: ##q## - magnitude of uniformly distributed load, ##a## - length of the plate (from fixed to free end), ##h## - thickness of the plate.
The results are in very good agreement with FEA (much better than when the plate is treated as a beam).
What are the edge boundary conditions for this solution?

Chestermiller said:
What are the edge boundary conditions for this solution?
One edge of the plate is fixed, the remaining ones are free:

FEAnalyst said:
One edge of the plate is fixed, the remaining ones are free:
View attachment 291921
On the constrained edge, is it constrained transversely so that the strains in the y direction are zero at this boundary?

Chestermiller said:
On the constrained edge, is it constrained transversely so that the strains in the y direction are zero at this boundary?
The boundary conditions are given in the book as follows: $$w_{x=a}=0$$ $$\varphi_{x=a}=\left( \frac{dw}{dx} \right)_{x=a}=0$$ $$t_{x=0}=-D \left( \frac{d^{3}w}{dx^{3}} \right)_{x=0}=0$$ $$\left( m_{y} \right)_{x=0}=-D \left( \frac{d^{2}w}{dx^{2}} \right)_{x=0}=0$$
where: ##t## - shear force, ##m## - bending moment.

## 1. What is the definition of "bending" in the context of a cantilever plate?

"Bending" refers to the deformation or curvature of a cantilever plate when a force is applied to it. This force causes the plate to bend or flex, rather than remain in its original flat position.

## 2. What factors affect the amount of bending in a cantilever plate?

The amount of bending in a cantilever plate is influenced by several factors, including the material properties of the plate (such as its stiffness and thickness), the magnitude and direction of the applied force, and the length and support conditions of the plate.

## 3. How is the bending moment calculated in a cantilever plate?

The bending moment in a cantilever plate is calculated by multiplying the applied force by the distance from the point of application to the fixed end of the plate. This can be represented mathematically as M = F x L, where M is the bending moment, F is the applied force, and L is the length of the plate.

## 4. What is the difference between a positive and negative bending moment in a cantilever plate?

A positive bending moment occurs when the top of the cantilever plate is in compression and the bottom is in tension, while a negative bending moment occurs when the top is in tension and the bottom is in compression. This can be visualized as the plate bending upwards or downwards, respectively.

## 5. How can the bending of a cantilever plate be minimized?

The bending of a cantilever plate can be minimized by using a stiffer and thicker material, reducing the applied force, and increasing the length of the plate. Additionally, adding supports at the fixed end of the plate can also help to reduce bending.

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